一类线性反应扩散方程最优控制问题的变步长BDF2格式的数值分析  

NUMERICAL ANALYSIS OF A CLASS OF LINEAR REACTION-DIFFUSION EQUATION OPTIMAL CONTROL PROBLEMS WITH VARIABLE STEP SIZE BDF2 SCHEME

作　　者：吕彤 叶星旸[1] Lyu Tong;Ye Xingyang(School of Science,Jimei University,Xiamen 361021,China)

机构地区：[1]集美大学理学院,厦门361021

出　　处：《计算数学》2025年第1期79-97,共19页Mathematica Numerica Sinica

基　　金：福建省自然科学基金面上项目(2022J01338,2024J01119);福建省高校数学学科联盟计划项目(2024SXLMMS03)资助。

摘　　要：时间变步长的两步向后差分公式(BDF2)具有强稳定性,在刚性问题、多尺度动力学等问题中具有广泛的应用,但在偏微分方程最优控制问题的应用研究相对较少.本文主要研究用变步长方法求解一类反应扩散方程源项控制的最优控制问题,时间方向采用变步长BDF2格式,空间方向采用中心差分方法进行离散.利用离散正交卷积(DOC)核和离散互补卷积(DCC)核的分析工具,证明了最优控制问题的最优解在相邻时间步长比介于1/4.8645和4.8645之间时,所构建的变步长差分格式在离散的L^(2)范数下是无条件稳定的,且在时间与空间方向都具有二阶收敛精度.最后通过数值算例验证了所构造格式的可行性和有效性.The Two-Step Backward Difference Formula(BDF2)of variable-step in time has ex-ceptional stability,making it an excellent choice for handling stif problems and multi-scale dynamics issues.However,there is limited research on the optimal control of Partial Differ-ential Equations.This paper introduces a variable-step method to solve the optimal control problem of source term control for a class of reaction-diffusion equations.Specifically,the BDF2 scheme is employed in time,while in space,we utilize the center-difference method for discretization.By utilizing the discrete orthogonal convolution(DOC)kernels and discrete complementary convolution(DCC)kernels,we establish the unconditional stability of our variable-step difference scheme in the L^(2)norm,provided that the ratio of adjacent time-steps fallswithin the range of a.sgas to 4.8645.Furthermore,it achieves second-order convergence accuracy in both time and space.Finally,two numerical examples are provided to validate the feasibility and effectiveness of the proposed scheme.

关 键 词：最优控制 变步长BDF2格式 稳定性 收敛性 

分 类 号：O232[理学—运筹学与控制论]